scholarly journals On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Representation Theory ◽  
Generating Function ◽  
Coxeter Group ◽  
Upper Bound ◽  
Open Problem ◽  
Special Class ◽  
Exact Enumeration ◽  
Restricted Permutations ◽  
International Audience ◽  
Freely Braided

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

scholarly journals The biHecke monoid of a finite Coxeter group

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Florent Hivert ◽  
Anne Schilling ◽  
Nicolas M. Thiéry
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Coxeter Group ◽  
Projective Modules ◽  
Permutation Matrices ◽  
Hecke Group ◽  
Finite Coxeter Group ◽  
Nous Montrons ◽  
Combinatorial Model ◽  
International Audience

arXiv : http://arxiv.org/abs/0912.2212 International audience For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0-Hecke$ algebra $H_0(W)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each $w∈W$ a combinatorial module $T_w$ whose support is the interval $[1,w]_R$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset. Pour tout groupe de Coxeter fini $W$, nous définissons deux nouveaux objets : son ordre de coupures et son monoïde de Hecke double. L'ordre de coupures, construit au moyen d'une généralisation de la notion de bloc dans les matrices de permutations, est presque un treillis sur $W$. La construction du monoïde de Hecke double s'appuie sur le modèle combinatoire usuel de la $0-algèbre$ de Hecke $H_0(W)$, pour le groupe symétrique, l'algèbre (ou le monoïde) engendré par les opérateurs de tri par bulles élémentaires. Les auteurs ont introduit précédemment l'algèbre de Hecke-groupe, construite comme l'algèbre engendrée conjointement par les opérateurs de tri et d'anti-tri, et décrit sa théorie des représentations. Dans cet article, nous considérons le monoïde engendré par ces opérateurs. Nous montrons qu'il admet $|W|$ modules simples et projectifs. Afin de construire ses modules simples, nous introduisons pour tout $w∈W$ un module combinatoire $T_w$ dont le support est l'intervalle [$1,w]_R$ pour l'ordre faible droit. Ce module détermine une algèbre dont la théorie des représentations généralise celle de l'algèbre de Hecke groupe, en remplaçant la combinatoire des descentes par celle des blocs et de l'ordre de coupures.

scholarly journals Enumerating projective reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Riccardo Biagioli ◽  
Fabrizio Caselli
Keyword(s):  
Representation Theory ◽  
Generating Functions ◽  
Special Class ◽  
Hilbert Series ◽  
Weyl Groups ◽  
Reflection Groups ◽  
One Dimensional ◽  
Complex Reflection Groups ◽  
Major Index ◽  
International Audience

International audience Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated. Les groupes de réflexions projectifs ont été récemment définis par le deuxième auteur. Ils comprennent une classe spéciale de groupes notée G(r,p,s,n), qui contient tous les groupes de Weyl classiques et plus généralement tous les groupes de réflexions complexes du type G(r,p,n). Dans ce papier on définit des statistiques analogues au nombre de descentes et à l'indice majeur pour les groupes G(r,p,s,n), et on calcule plusieurs fonctions génératrices. Certains aspects de la théorie des représentations de G(r,p,s,n), comme la distribution des caractères linéaires et le calcul de la série de Hilbert de quelques algèbres d'invariants, sont aussi abordés.

scholarly journals New type of degenerate Daehee polynomials of the second kind

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Serkan Araci ◽  
Sameh S. Ahmed
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials ◽  
New Type ◽  
Order Degenerate ◽  
Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

scholarly journals Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

2018 ◽  
Vol 28 (3) ◽  
pp. 365-387
Author(s):  
S. CANNON ◽  
D. A. LEVIN ◽  
A. STAUFFER
Keyword(s):  
Markov Chain ◽  
Relaxation Time ◽  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Mixing Time ◽  
Perpendicular Bisector ◽  
Unit Square

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

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